Calculus Examples

$\int \arccos (2 x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \arccos (2 x)$ and $d v = 1$ .

$\arccos (2 x) x - \int x (- \frac{2}{\sqrt{1 - 4 x^{2}}}) d x$

Step 2

Simplify.

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Step 2.1

Combine $x$ and $\frac{2}{\sqrt{1 - 4 x^{2}}}$ .

$\arccos (2 x) x - \int - \frac{x \cdot 2}{\sqrt{1 - 4 x^{2}}} d x$

Step 2.2

Move $2$ to the left of $x$ .

$\arccos (2 x) x - \int - \frac{2 x}{\sqrt{1 - 4 x^{2}}} d x$

Step 3

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\arccos (2 x) x - - \int \frac{2 x}{\sqrt{1 - 4 x^{2}}} d x$

Step 4

Simplify.

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Step 4.1

Multiply $- 1$ by $- 1$ .

$\arccos (2 x) x + 1 \int \frac{2 x}{\sqrt{1 - 4 x^{2}}} d x$

Step 4.2

Multiply $\int \frac{2 x}{\sqrt{1 - 4 x^{2}}} d x$ by $1$ .

$\arccos (2 x) x + \int \frac{2 x}{\sqrt{1 - 4 x^{2}}} d x$

Step 5

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$\arccos (2 x) x + 2 \int \frac{x}{\sqrt{1 - 4 x^{2}}} d x$

Step 6

Let $u = 1 - 4 x^{2}$ . Then $d u = - 8 x d x$ , so $- \frac{1}{8} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.1

Let $u = 1 - 4 x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 6.1.1

Differentiate $1 - 4 x^{2}$ .

$\frac{d}{d x} [1 - 4 x^{2}]$

Step 6.1.2

Differentiate.

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Step 6.1.2.1

By the Sum Rule, the derivative of $1 - 4 x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 4 x^{2}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [- 4 x^{2}]$

Step 6.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 4 x^{2}]$

Step 6.1.3

Evaluate $\frac{d}{d x} [- 4 x^{2}]$ .

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Step 6.1.3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{2}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{2}]$ .

$0 - 4 \frac{d}{d x} [x^{2}]$

Step 6.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0 - 4 (2 x)$

Step 6.1.3.3

Multiply $2$ by $- 4$ .

$0 - 8 x$

Step 6.1.4

Subtract $8 x$ from $0$ .

$- 8 x$

Step 6.2

Rewrite the problem using $u$ and $d u$ .

$\arccos (2 x) x + 2 \int \frac{1}{\sqrt{u}} \cdot \frac{1}{- 8} d u$

Step 7

Simplify.

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Step 7.1

Move the negative in front of the fraction.

$\arccos (2 x) x + 2 \int \frac{1}{\sqrt{u}} (- \frac{1}{8}) d u$

Step 7.2

Multiply $\frac{1}{\sqrt{u}}$ by $\frac{1}{8}$ .

$\arccos (2 x) x + 2 \int - \frac{1}{\sqrt{u} \cdot 8} d u$

Step 7.3

Move $8$ to the left of $\sqrt{u}$ .

$\arccos (2 x) x + 2 \int - \frac{1}{8 \sqrt{u}} d u$

Step 8

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$\arccos (2 x) x + 2 (- \int \frac{1}{8 \sqrt{u}} d u)$

Step 9

Multiply $- 1$ by $2$ .

$\arccos (2 x) x - 2 \int \frac{1}{8 \sqrt{u}} d u$

Step 10

Since $\frac{1}{8}$ is constant with respect to $u$ , move $\frac{1}{8}$ out of the integral.

$\arccos (2 x) x - 2 (\frac{1}{8} \int \frac{1}{\sqrt{u}} d u)$

Step 11

Simplify the expression.

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Step 11.1

Simplify.

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Step 11.1.1

Combine $\frac{1}{8}$ and $- 2$ .

$\arccos (2 x) x + \frac{- 2}{8} \int \frac{1}{\sqrt{u}} d u$

Step 11.1.2

Cancel the common factor of $- 2$ and $8$ .

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Step 11.1.2.1

Factor $2$ out of $- 2$ .

$\arccos (2 x) x + \frac{2 (- 1)}{8} \int \frac{1}{\sqrt{u}} d u$

Step 11.1.2.2

Cancel the common factors.

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Step 11.1.2.2.1

Factor $2$ out of $8$ .

$\arccos (2 x) x + \frac{2 \cdot - 1}{2 \cdot 4} \int \frac{1}{\sqrt{u}} d u$

Step 11.1.2.2.2

Cancel the common factor.

$\arccos (2 x) x + \frac{2 \cdot - 1}{2 \cdot 4} \int \frac{1}{\sqrt{u}} d u$

Step 11.1.2.2.3

Rewrite the expression.

$\arccos (2 x) x + \frac{- 1}{4} \int \frac{1}{\sqrt{u}} d u$

Step 11.1.3

Move the negative in front of the fraction.

$\arccos (2 x) x - \frac{1}{4} \int \frac{1}{\sqrt{u}} d u$

Step 11.2

Apply basic rules of exponents.

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Step 11.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$\arccos (2 x) x - \frac{1}{4} \int \frac{1}{u^{\frac{1}{2}}} d u$

Step 11.2.2

Move $u^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\arccos (2 x) x - \frac{1}{4} \int {(u^{\frac{1}{2}})}^{- 1} d u$

Step 11.2.3

Multiply the exponents in ${(u^{\frac{1}{2}})}^{- 1}$ .

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Step 11.2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\arccos (2 x) x - \frac{1}{4} \int u^{\frac{1}{2} \cdot - 1} d u$

Step 11.2.3.2

Combine $\frac{1}{2}$ and $- 1$ .

$\arccos (2 x) x - \frac{1}{4} \int u^{\frac{- 1}{2}} d u$

Step 11.2.3.3

Move the negative in front of the fraction.

$\arccos (2 x) x - \frac{1}{4} \int u^{- \frac{1}{2}} d u$

Step 12

By the Power Rule, the integral of $u^{- \frac{1}{2}}$ with respect to $u$ is $2 u^{\frac{1}{2}}$ .

$\arccos (2 x) x - \frac{1}{4} (2 u^{\frac{1}{2}} + C)$

Step 13

Simplify.

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Step 13.1

Rewrite $\arccos (2 x) x - \frac{1}{4} (2 u^{\frac{1}{2}} + C)$ as $\arccos (2 x) x + \frac{- u^{\frac{1}{2}}}{2} + C$ .

$\arccos (2 x) x + \frac{- u^{\frac{1}{2}}}{2} + C$

Step 13.2

Move the negative in front of the fraction.

$\arccos (2 x) x - \frac{u^{\frac{1}{2}}}{2} + C$

Step 14

Replace all occurrences of $u$ with $1 - 4 x^{2}$ .

$\arccos (2 x) x - \frac{{(1 - 4 x^{2})}^{\frac{1}{2}}}{2} + C$

Step 15

Reorder terms.

$\arccos (2 x) x - \frac{1}{2} {(1 - 4 x^{2})}^{\frac{1}{2}} + C$